JHEP04(2016)172

Published for SISSA by Springer

Received: January 30, 2016

Revised: March 10, 2016

Accepted: April 21, 2016

Published: April 28, 2016

D1 string dynamics in curved backgrounds with fluxes

Aritra Banerjee,a Sagar Biswasb and Rashmi R. Nayaka

aDepartment of Physics, Indian Institute of Technology Kharagpur,

Kharagpur-721 302, IndiabDepartment of Physics, R.K.M. Vidyamandira,

Belur Math, Howrah 711 202, India

E-mail: aritra@phy.iitkgp.ernet.in, biswas.sagar09iitkgp@gmail.com,

rashmi.string@gmail.com

Abstract: We study various rotating and oscillating D-string configurations in some gen-

eral backgrounds with fluxes. In particular, we look for solutions to the equations of motion

of various rigidly rotating D-strings in AdS3 background with mixed flux, and in the in-

tersecting D-brane geometries. We find out relations among various conserved charges

corresponding to the breathing and rotating D-string configurations.

Keywords: AdS-CFT Correspondence, Bosonic Strings, D-branes

ArXiv ePrint: 1601.06360

Open Access, c© The Authors.

Article funded by SCOAP3.doi:10.1007/JHEP04(2016)172

JHEP04(2016)172

Contents

1 Introduction and summary 1

2 Circular (m,n) strings on AdS3 with mixed 3-form fluxes 3

3 Rotating D1-string on D5-branes 7

3.1 Single spike solution 10

3.2 Giant Magnon 13

4 Rotating D1-strings on intersecting D5-D5’ branes 14

4.1 Single Spike-like solution 18

4.2 Giant Magnon-like solution 20

5 Conclusions 22

1 Introduction and summary

The AdS/CFT correspondence [1–3], relates string states in AdS spaces to some conformal

field theory (CFT) living on its boundary. In general, it is very difficult to find the full

spectrum of states in the string side and then compare it with the anomalous dimensions of

the operators on the CFT side. This has been tough even for the very well studied example

of the N = 4 supersymmetric Yang-Mills (SYM) theory in four dimensions and dual type

IIB superstring in the compactified AdS5 space. One of the direct ways to check the

correspondence beyond the supergravity approximation has been to study various classical

string solutions in varieties of target space geometries and using the dispersion relation

of such strings in the large charge limit, one could look for boundary operators dual to

them.1 This has been one of the main ways to establish the AdS/CFT dictionary in

many cases. Instead of probe classical fundamental strings, there have also been many

studies on various exact string backgrounds using probe Dp-branes. This provides a novel

way of understanding string theory in curved background as the Dp-brane couples to RR

fluxes present.

One of the first attempts in this direction was to study D-branes in the SL(2,R)

Wess-Zumino-Witten (WZW) model and its discreet orbifolds [5]. The corresponding tar-

get space geometries are mainly AdS3 with non-trivial NS-NS fluxes and 3d AdS black

holes [6, 7]. This was extensively studied in a lot of works and the WZW D-branes are now

well understood from both CFT boundary states and the target space viewpoint [8–15].

These brane worldvolumes are also shown to carry non-trivial gauge fields. Also coupling to

background NS-NS fluxes resists a D-string probe from reducing to a point particle [13, 14].

1For a review of semiclassical strings in AdS/CFT one may look at [4].

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JHEP04(2016)172

However, it was also shown that under a supercritical worldvolume electric field, a circu-

lar oscillating D-string can never reach the boundary of AdS space [16]. In a related

development, D1 strings rotating in R × S2 and R × S3 were studied in detail by using

Dirac-Born-Infeld (DBI) action [17] to find giant magnon [18] and single spike [19] like

solutions on the D1-string. Giant magnons arise as rotating solutions in the string side

corresponding to low lying spin-chain excitations in the dual field theory. Similarly single

spike configurations are particular string solutions which correspond to a particular class

of single trace operators in the field theory with large number of derivatives. Both of these

appear generally as fundamental string solutions and the D1-string analog in the presence

of worldvolume gauge field indeed appears to have yet-unknown novel interpretation.

In general, D1 string solutions in Dp-brane backgrounds are rare due to the complicated

nature of the background and presence of dilaton and other RR fields. But this problem

appears to be interesting in conjunction to the prediction of [20] that a general Dp-brane

background is non-integrable for extended objects. However various well behaved probe

brane solutions have been constructed in these backgrounds too. One example is in [21, 22],

where the system consisting of two stacks of the fivebranes in type IIB theory that intersect

on R1,1 (an Intersecting brane or I-brane [23]) has been investigated using probe D-strings.

The much discussed enhancement of symmetry in the near horizon geometry of such systems

have been shown to have profound impact on the D1-string worldvolume itself.

Motivated by the above studies, we move on to discuss probe D1 string solutions in

a few general settings with coupling to background fluxes. The first study we undertake

is a natural generalization of the WZW D-brane solutions. Recently, it was shown that

the string theory on AdS3 × S3 supported by both NS-NS and RR three-form fluxes is

integrable [24, 25]. There has been proposals of S-matrix on this background [26–32]

and various classical string solutions [29, 33–39] have been constructed. The NS-NS flux

in this background is parameterized by a number q with 0 ≤ q ≤ 1, while the RR 3-

form is proportional to q =√

1− q2. As the q interpolates between 0 to 1, the solution

interpolates between that of a pure RR background and a pure NS-NS backgound described

by the usual WZW model. However, for intermediate values of q, the exact description of

the string sigma model is not known. In any case the study of open string integrability

in this background is still lacking, most probably due to dearth of D-brane boundary

conditions. However, recently the integrability of a D1-string on the group manifold with

mixed three form fluxes has been argued and Lax connections have been constructed [40]

with the condition that dilaton and Ramond-Ramond zero forms are constants. It is worth

noting that the analysis has only been performed at the bosonic level and it remains to

show whether the full action is integrable. Still, even in the bosonic sector, it remains an

interesting problem to analyse dynamics of D1-strings in this background. We will probe

this backgound with a bound state of oscillating D1 strings and F-strings with non-trivial

gauge field on the D1 worldvolume. One of the apparent outcomes of the analysis based

on the DBI action is that the periodically expanding and contracting (1, n) string has a

possibility of reaching the boundary of AdS3 in finite time in contrast to the probe D-string

motion in the WZW model with only NS-NS fluxes.

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JHEP04(2016)172

The later parts of the paper is devoted to study of rotating D1 string solutions in

various Dp brane backgrounds. In particular we will talk about the near horizon geometry

of a stack of D5 branes and two stacks of D5 branes that intersect on a line. We solve

the equations of motion for D1 string keeping couplings with dilatons and WZ terms in

mind. It is shown that in proper limits, the combination of properly regularised conserved

quantities actually give rise to giant magnon and single spike like dispersion relations as

in the case of fundamental strings. We speculate that these exact solutions correspond to

S-dual fundamental string solutions in NS5 brane background [41] and in NS5-NS5’ brane

intersections [42].

The rest of the paper is organised as follows. In section 2 we will discuss the motion

of a (m,n) string in the mixed-flux AdS3 background. In section 3, we will move on to

the motion of a rotating D1-string in the near horizon geometry of a stack of D5 branes.

We will show that the conserved charges give rise to giant magnon or single spike-like

dispersion relations in two different limits. We will discuss a similar configuration in the

intersecting D5-D5’ brane background in section 4. The equations of motion appear to be

very complicated in that case, but with certain simplifications we will be again able to find

giant magnon or single spike-like dispersion relations for the properly regularised version

of the charges. We conclude with some outlook in section 5.

2 Circular (m,n) strings on AdS3 with mixed 3-form fluxes

The mixed flux background is a solution of the type IIB action with a AdS3 × S3 × T 4

geometry, although the compact manifold will not be interesting here. The solution has

both RR and NS-NS fluxes along the AdS and S directions. In this section we will put the

S3 coordinates to be constant and consider motion only along the AdS3. The background

and the relevant fluxes are as follows,

ds2 = − cosh2 ρdt2 + dρ2 + sinh2 ρdφ2 ,

B(2) = q cosh2 ρ dt ∧ dφ , C(2) =√

1− q2 cosh2 ρ dt ∧ dφ . (2.1)

The dilaton Φ is constant and can be set to zero. Also, the AdS radius has been properly

chosen here. The above background can easily be shown to be a solution of the type IIB

field equations.2

Now, we want to discuss the motion of a bound state of m D1-strings and n F-strings

in this background as a general string state in type IIB theory would readily be the bound

state of the two. For simplicity we will choose m = 1 and argue that the analysis can be

extended to other more general string state too. The DBI action for a D1-brane can be

written as,

S = −TD

∫

d2ξe−φ√

− det(g + B + 2πα′F ) +TD2

∫

d2ξǫαβCαβ . (2.2)

2Note that there is a gauge freedom in choosing the two-form B-field since the supergravity equations of

motion involves H(3) = dB(2) only. For example, in the case of a three sphere the NS-NS flux is defined upto

an additive constant in the form − q

2(cos 2θ + c). This constant can be fixed using physical considerations.

We have fixed the constant here for the AdS3 case so that the fields have the mentioned form. For details

one could see the arguments presented in [29].

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JHEP04(2016)172

where g and B are the pullback of the background metric and the NS-NS fluxes and F

is the field strength for the worldvolume gauge field. Here TD is the D1-string tension

TD = 1/(2πα′gs), with gs being the string coupling. In contrast, remember that the F-

string tension was simply TF = 1/2πα′. So in weak coupling regime we can write TD ≫ TF .

Also, ǫαβ is the usual antisymmetric tensor with ǫ01 = 1. Since the mixed flux background

has the constant (or zero) dilaton, the Lagrangian density takes the form,

L = −TD

[

√

− det(g + B + 2πα′F )− 1

2ǫαβCαβ

]

. (2.3)

We choose the following ansatz for a ‘breathing’ mode of the string:

t = ξ0 , ρ(ξ0, ξ1) = ρ(ξ0) , φ = ξ1 . (2.4)

The ansatz is like the one for a simple circular F-string motion and can be easliy shown to

be consistent with the equations of motion here. Using the above ansatz we can show,

− det(g + B + 2πα′F ) = sinh2 ρ(cosh2 ρ− (∂0ρ)2)− (q cosh2 ρ− 2πα′∂0Aφ)

2

= − det g −F2φt , (2.5)

where we have

det g = − sinh2 ρ(cosh2 ρ− (∂0ρ)2) (2.6)

and

Fφt = q cosh2 ρ− 2πα′∂0Aφ. (2.7)

Now, we can rewrite the lagrangian density as,

L = −TD

[√

− det g −F2φt −

√

1− q2 cosh2 ρ]

. (2.8)

Conjugate momentum of the Wilson line Aφ is a quantized constant of the motion given by,

1

2πΠφ =

∂L∂(∂0Aφ)

=−2πα′TDFφt

√

− det g −F2φt

= −n . (2.9)

The integer n is the number of oriented fundamental strings bound to the D-string. The

effective tension of (m,n) circular string is given by,

T(m,n) =√

m2T 2D + n2T 2

F . (2.10)

From the above expression and (2.9) we can write,

T(1,n) = TD

√

√

√

√1 +(2πα′)2T 2

FF2φt

(− det g −F2φt)

. (2.11)

Combining all expression we get the following relation which we will later use to simplify

other expressions,T(1,n)√− det g

=TD

√

− det g −F2φt

=n

2πα′Fφt. (2.12)

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JHEP04(2016)172

The other constant of the motion is the energy, as measured by an observer sitting at the

center of AdS3,

E = 2π

(

∂L∂(∂0ρ)

∂0ρ+∂L

∂(∂0Aφ)∂0Aφ − L

)

= 2πTD

sinh2 ρ cosh2 ρ−q cosh2 ρ(q cosh2 ρ−2πα′∂0Aφ)√

− det g −F2φt

−√

1−q2 cosh2 ρ

. (2.13)

Putting the energy in a more suggestive form we can get,

E =2πT(1,n) sinh ρ cosh

2 ρ√

cosh2 ρ− (∂0ρ)2− 2π cosh2 ρ

(

nqTF +√

1− q2TD

)

. (2.14)

Above equation exhibits the competing terms of the potential energy: the blue-shifted

mass, and the interaction with the B-field and C-field potential. Using the above we can

write the equation of motion for ρ as,

∂0ρ =cosh ρ

E + C2 cosh2 ρ

[

(E + C2 cosh2 ρ)2 − C2

1 sinh2 ρ cosh2 ρ

]1/2(2.15)

Where the constants

C1 = 2πT(1,n) C2 = 2π(

nqTF +√

1− q2TD

)

(2.16)

Note that C1 > C2 is still valid as we have q ∈ [0, 1]. Now our interest is to find a large

energy solution, i.e. a limit where the string becomes ‘long’ and tries to reach the boundary

of AdS in finite time. In this limit we can solve the radial equation perturbatively in orders

of 1E . Expanding (2.15), we get,

∂0ρ = cosh ρ− C21 sinh

2 ρ cosh3 ρ

2E2+

C21C2 sinh

2 ρ cosh5 ρ

E3+O

(

1

E4

)

(2.17)

To solve this order by order we take the expression for ρ

ρ = ρ(0) +ρ(1)

E2+

ρ(2)

E3+O

(

1

E4

)

(2.18)

Putting the above back in (2.15) and expanding we get the following set of coupled differ-

ential equations for the different orders

∂0ρ(0) = cosh ρ(0),

∂0ρ(1) = ρ(1) sinh ρ(0) − 1

2C21 cosh

3 ρ(0) sinh2 ρ(0),

∂0ρ(2) = ρ(2) sinh ρ(0) + C2

1C2 cosh5 ρ(0) sinh2 ρ(0) .

(2.19)

We solve the first equation to find

ρ(0) = sinh−1 tan τ, (2.20)

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JHEP04(2016)172

which is a periodically expanding and contracting solution where the string goes out to a

maximum radius. We use this solution iteratively in the other two equations and find the

total solution using the boundary condition ρ(0) = 0 to write the perturbative solution

ρ(τ) = sinh−1 tan τ − 1

6E2C21 sec τ tan

3 τ+1

15E3C21C2(4+cos 2τ) tan3 τ sec3 τ+O

(

1

E4

)

(2.21)

Now the dynamics is incredibly difficult as there are many parameters involved like TF ,

TD, n and q. The dynamics varies widely for different values of these parameters. With

higher order corrections (which are suppressed in the large energy limit), the (1, n) string

goes through quasi-periodic motion.

Now to find the maximum radius of the circular D-string we note that the extremum

would occur at ∂0ρ = 0, which leads us to the following,

E + C2 cosh2 ρm − C1 sinh ρm cosh ρm = 0. (2.22)

With a little algebra we can find that,

ρm =1

2ln

[

2E + C2 ±√

4E2 + 2EC2 + C21

(C1 − C2)

]

. (2.23)

To prove that the above corresponds to the maximum value of ρ we can explicitly show that

∂20ρ(ρm) < 0. (2.24)

Since we are interested in the large E limit of the solution, we can write the maximum

radius approximately in a simpler form by putting in the expressions for C1,2,

ρm ≃ 1

2ln

[

2E

π(T(1,n) − nqTF −√

1− q2TD)

]

(2.25)

If one keeps all other parameters fixed and increases q from 0, it can easily be seen that the

maximum value of ρ actually increases. So, we can say that with increasing NS-NS flux

the large energy string actually gets ‘fatter’. However, we are interested to know whether

the string becomes a ‘long’ one. Now interestingly enough, one can see that the expression

for maximum radius diverges when

T(1,n) → nqTF +√

1− q2TD. (2.26)

Note that diverging ρm means that the strings can actually expand upto the boundary of the

AdS3. For the case of pure NS-NS flux i.e. q = 1 as discussed in [16], the ρm can only diverge

if T(1,n) → nTF , so that the string becomes a purely fundamental ‘long’ string. It was made

clear in [16] that a general (m,n) string can never reach the boundary of AdS as the first

term in (2.14) diverges faster near the boundary than the NS-NS flux potential term for

q = 1, making the energy effectively infinite. It is quite clear that for pure NS-NS case the

two terms in (2.14) would cancel in the asymptotic region to produce a finite contribution,

only if the string is purely fundamental. However here we also have contribution from the

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JHEP04(2016)172

RR term, so the string might not need to be purely fundamental to acquire a finite energy

near the boundary. In our example, keeping in mind that T(1,n) =√

T 2D + n2T 2

F , we can

easily see the following,

TD + nTF > T(1,n),

and TD + nTF > nqTF +√

1− q2TD; 0 < q < 1. (2.27)

So in any case the condition (2.26) actually could be physical in a particular region of

the parameter space and the boundary might not remain forbidden region for the (1, n)

string when it couples to both NS-NS and RR fluxes. In fact if we impose an equality in

the (2.26), it simply yields the condition

n√

1− q2 TF = ±qTD, (2.28)

for the maximum radius of the string to diverge. We shall not be bothering about the

negative sign here. It is very clear that in the weak coupling region (small gs), we can

transform the above condition into,

n

√

1− q2

q2≫ 1. (2.29)

As we have n to be a positive finite integer, the above condition can only be satisfied if the

value of q is small. As q → 1, the above inequality is violated since the equality associated

with (2.26) loses meaning for this case and can only be true if the string becomes a purely

fundamental one. This supports the discussion we presented earlier. This is a unique

observation for such strings in this mixed flux background and have to be investigated into

details using other approaches.

3 Rotating D1-string on D5-branes

Let us first start with the discussion of generic Dp brane backgrounds. For an arbitrary p,

we can write the general supergravity solution as follows,

ds2 = h−12 (~x)(−dt2 + d~y 2) + h

12 (~x) d~x 2,

e2φ = h(~x)3−p

2 ,

Cp+1 = ±(

1

h(~x)− 1

)

dt ∧ dy1 ∧ . . . ∧ dyp,

F p+2 = dCp+1 = ∓h−2∂jh dxj ∧ dt ∧ dy1 ∧ . . . ∧ dyp,

F 8−p = ±∂jh ixj (dx1 ∧ . . . ∧ dx9−p),

h(~x) = 1 +µ

7− p

(

lsr

)7−p

. (3.1)

Let us remind ourselves the various expressions used in the above equation. Here p spatial

coordinates yk run parallel to the worldvolume while the transverse space is given by (9−p)

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JHEP04(2016)172

coordinates labelled by xi. The specified harmonic function h(~x) is valid for p = 0, 1, 2 . . . , 6

with r2 =∑(9−p)

i=1 xixi being the radial distance along the transverse space. The dilaton is

given by φ and the RR form is given by Cp+1. The RR field strength is given by F p+2 while

the hodge dual field strength is given by F 8−p. The action of ixj on the volume form of the

transverse space signifies a inner product with a unit vector pointing in the xj direction.

For a D1-string moving in a general Dp-brane background, the DBI action is given by,

S = −T1

∫

dξ0dξ1e−φ√

− detAαβ + T1

∫

dξ0dξ1C01 (3.2)

where Aαβ = gαβ = ∂αXM∂βX

NgMN is the induced metric on the worldvolume and

α, β = ξ0, ξ1. We here put the worldvolume gauge field to be zero for simplicity. C01 is the

two form RR potential coupled to the D1 string.

The background in which we are interested in is a stack of N D5-branes, which is given

by the following metric, dilaton and RR six-form:

ds2 = h−12

[

−dt2 +5

∑

i=1

dx2i

]

+ h12

[

dr2 + r2(dθ2 + sin2 θdφ21 + cos2 θdφ2

2)]

eφ = h−12 , C012345 =

k

k + r2, h = 1 +

k

r2, k = gsNl2s . (3.3)

Here ls is the string length scale and N is the number of branes in the stack. In the

near horizon limit r → 0, so in the harmonic function, the second term dominates and it

becomes h = 1 + kr2

≈ kr2

and also the six form potential C012345 = 1 + O(r2). We now

rescale the coordinates in the following way

t →√kt xi →

√kxi, (3.4)

and following this, the metric and fields reduce to the form,

ds2 =√kr

[

−dt2 +5

∑

i=1

dx2i +dr2

r2+ dθ2 + sin2 θdφ2

1 + cos2 θdφ22

]

eφ =r√k, C012345 = 1, k = gsNl2s . (3.5)

This six form background RR field does not couple to a D1-string. However, the dual of

this six form i.e. the ‘magnetic’ two form RR field (Cφ1φ2) will couple to the of D1-string

as WZ contribution.3 For our convenience we define ρ = ln r, and transform the the metric

and the dilaton to,

ds2 =√keρ

[

−dt2 +5

∑

i=1

dx2i + dρ2 + dθ2 + sin2 θdφ21 + cos2 θdφ2

2

]

eφ =eρ√k, Cφ1φ2 = 2k sin2 θ, k = gsNl2s . (3.6)

3For detailed discussion on this point one could see the discussion in [43].

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JHEP04(2016)172

Now, it is to be noted that the RR two form (Cφ1φ2) has been written upto an additive

constant as we can calculate the field strength F (3) from the (3.1). This 3-form turns out

simply to be the volume form of the transverse sphere itself. Now for our probe D1 string,

the induced worldvolume metric components are given by,

A00 =√keρ

[

− (∂0t)2 +

∑

(∂0xi)2 + (∂0ρ)

2 + (∂0θ)2 + sin2 θ(∂0φ1)

2 + cos2 θ(∂0φ2)2]

A11 =√keρ

[

− (∂1t)2 +

∑

(∂1xi)2 + (∂1ρ)

2 + (∂1θ)2 + sin2 θ(∂1φ1)

2 + cos2 θ(∂1φ2)2]

A01 = A10 =√keρ

[

− (∂0t)(∂1t) +∑

(∂0xi)(∂1xi) + (∂0ρ)(∂1ρ) + (∂0θ)(∂1θ)

+ sin2 θ(∂0φ1)(∂1φ1) + cos2 θ(∂0φ2)(∂1φ2)]

. (3.7)

Therefore we can write the total Lagrangian-density as,

L = −T1e−φ

√

− detAαβ +T1

2ǫαβ∂αX

M∂βXNCMN

= −T1k

[

[

− (∂0t)(∂1t) +∑

(∂0xi)(∂1xi) + (∂0ρ)(∂1ρ) + (∂0θ)(∂1θ)

+ sin2 θ(∂0φ1)(∂1φ1) + cos2 θ(∂0φ2)(∂1φ2)]2

−[

− (∂0t)2 +

∑

(∂0xi)2

+ (∂0ρ)2 + (∂0θ)

2 + sin2 θ(∂0φ1)2 + cos2 θ(∂0φ2)

2][

− (∂1t)2

+∑

(∂1xi)2 + (∂1ρ)

2 + (∂1θ)2 + sin2 θ(∂1φ1)

2 + cos2 θ(∂1φ2)2]

]12

+ 2T1k sin2 θ(∂0φ1∂1φ2 − ∂0φ2∂1φ1) . (3.8)

Before solving the Euler-Lagrange equations,

∂0

(

∂L∂(∂0X)

)

+ ∂1

(

∂L∂(∂1X)

)

=∂L∂X

, (3.9)

we choose a rotating ansatz for the probe brane,

t = κξ0, xi = νiξ0, i = 1, 2, 3, 4, 5, ρ = mξ0,

θ = θ(ξ1), φ1 = ω1ξ0 + ξ1, φ2 = ω2ξ

0 + φ2(ξ1) . (3.10)

The above ansatz can be thought of as the generalized version of the rigidly rotating string

parameterisation described in, for example, [19]. Solving the equations of motion for φ1

and φ2 and eliminating L from the respective equations we get,

∂φ2

∂ξ1=

sin2 θ[(c2−2ω2 sin2 θ)ω1ω2 cos

2 θ+(c3+2ω1 sin2 θ)ω2

2 cos2 θ−α2(c3+2ω1 sin

2 θ)]

cos2 θ[(c3+2ω1 sin2 θ)ω1ω2 sin

2 θ+(c2−2ω2 sin2 θ)ω2

1 sin2 θ−α2(c2−2ω2 sin

2 θ)].

(3.11)

Again solving for t, we get the equation for θ,

( ∂θ

∂ξ1

)2=

(c21 − κ2)(ω1 sin2 θ + ω2∂1φ2 cos

2 θ)2

c21−α2 + ω21 sin

2 θ + ω22 cos

2 θ− sin2 θ + (∂1φ2)

2 cos2 θ . (3.12)

where c1, c2 and c3 are carefully chosen integration constants and α2 = κ2−m2−∑5i=1 νiν

i.

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JHEP04(2016)172

Certainly, the equation of motion for θ is quite complicated and difficult to solve in

general due to the presence of all kinds of constants. In what follows we will discuss

two limiting cases from the above set of equations corresponding to single spike and giant

magnon solutions. In [19] it was shown that the giant magnon and single spike strings can

be thought of two limits of the same system of equations. We will show that this is the

same for our case also.

Let us outline the procedure to impose conditions on the equations of motion in a

simple schematic way. The main problem lies in the behaviour of ∂φ2∂ξ1

at θ = π2 as there is

a cos2 θ sitting in the denominator. To talk about the behaviour of ∂θ∂ξ1

in this limit, we

will then define(

∂φ2

∂ξ1

)2

cos2 θ

∣

∣

∣

∣

θ=π2

= P2. (3.13)

Now we want to have particular conditions imposed on ∂θ∂ξ1

too. We can see that

∂θ

∂ξ1

∣

∣

∣

∣

θ=π2

=c21α

2 − κ2ω21 + P2c21(α

2 − ω21)

c21(−α2 + ω21)

. (3.14)

There are two very interesting limits that can be considered here. The first one is to

demand that ∂θ∂ξ1

→ 0 as θ → π2 , which in turn demands that c1α = κω1. Also it is

required in this case that P2 → 0 i.e. ∂φ2∂ξ1

has to be regular, which in turn demands that

c3 = −2ω1. So, this set of relations between the constants will implement the conditions

mentioned above. The other limit may be taken as ∂θ∂ξ1

→ ∞ as θ → π2 . It also demands

that ∂φ2∂ξ1

→ ∞ in the same limit of θ. Both of these can be realised by the set of relations

α = ω1 and c3 = −2ω1. We will discuss these particular two conditions while finding out

the constants of motion and the relevant dispersion relations among them. We will see

that the equations of motion considerably simplify when we put in the relations between

the constants.

3.1 Single spike solution

In this section we will follow the procedure outlined in [19] and consider constants of

motions appropriately, demanding the following condition is satisfied,

∂θ

∂ξ1→ 0 as θ → π

2. (3.15)

As we discussed earlier, this will also require ∂φ2∂ξ1

to be regular. So combining these,

we get the relations between the constants c1α = κω1 and c3 = −2ω1. Note that the

imposed conditions determine the value of integration constants c1 and c3 in terms of other

parameters, but it doesn’t fix the value of c2. For the time being we can keep c2 as it

is. Later, we will try to fix a value of c2 using other considerations. Using these the

equations (3.11) and (3.12) transform to,

∂φ2

∂ξ1=

ω1(2ω22 − 2α2 − c2ω2) sin

2 θ

c2α2 + (2ω21ω2 − 2α2ω2 − c2ω2

1) sin2 θ

, (3.16)

– 10 –

JHEP04(2016)172

and

∂θ

∂ξ1=

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2 sin θ cos θ√

sin2 θ − sin2 θ0

c2α2 + (2ω21ω2 − 2α2ω2 − c2ω2

1) sin2 θ

. (3.17)

Where we have the upper limit on θ in the form,

sin θ0 =c2α

√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2. (3.18)

Now we can explicitly solve for θ equation from (3.17) to get the profile of the string, which

gives

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2 ξ1 =c2(α

2 − ω21) + (2ω2

1ω2 − 2α2ω2)

cos θ0cosh−1

[

cos θ0cos θ

]

− c2α2

sin θ0cos−1

[

sin θ0sin θ

]

. (3.19)

From the above we can see θ = θ0 can be thought of as the point exactly where the tip

of the spike is situated. On the other hand it is clear that at θ → π2 we actually have

ξ1 → ∞, making this a valid single spike solution. The largest spike goes up to the pole

of the sphere where θ0 = 0. Now we can calculate the conserved charges for the D-string

motion using usual noether techniques,

E = −2

∫ θ1

θ0

∂L∂0t

dθ

∂1θ=

2T1kκ(α2 − ω2

1)(c2 − 2ω2)

α2√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0,

Pi = 2

∫ θ1

θ0

∂L∂0xi

dθ

∂1θ=

2T1kνi(α2 − ω2

1)(c2 − 2ω2)

α2√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0,

D = 2

∫ θ1

θ0

∂L∂0ρ

dθ

∂1θ=

2T1km(α2 − ω21)(c2 − 2ω2)

α2√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0.

(3.20)

Here as usual, the E corresponds to time translation invariance, and Pi’s correspond to

shifts in xi’s. The other charge D comes from translations in ρ = ln r. In general this

is not a symmetry of the metric itself, but the way we have chosen the parameterisations

in (3.10) makes it clear that these are indeed symmetries of the D1 string action. All these

quantities diverge due to the divergent integral. The angular momenta J1 and J2 that arise

from the isometries along φ1 and φ2 are given by,

J1 =2T1kω1[2(2ω

22 − 2α2 − c2ω2)− c2α]

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θ cos θdθ√

sin2 θ − sin2 θ0

− 4T1kω1[(2ω22 − 2α2 − c2ω2)]

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0. (3.21)

J2 =2T1k[α(2ω

21 − 2ω2

2 + c2ω2)− 2(2ω21ω2 − 2α2ω2 − c2ω

21)]

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θ cos θdθ√

sin2 θ − sin2 θ0

+2T1k[2(c2α

2 + 2ω21ω2 − 2α2ω2 − c2ω

21)]

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0. (3.22)

– 11 –

JHEP04(2016)172

The other quantity of interest is the angle deficit defined as ∆φ = 2∫ θ1θ0

dθ∂1θ

is given by,

∆φ =2

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2

[

(c2 − 2ω2)(α2 − ω2

1)

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0

+ c2α2

∫ π2

θ0

cos θ

sin θ√

sin2 θ − sin2 θ0

]

, (3.23)

which is clearly divergent because of the first integral. However one can regularize the angle

difference using a combination of other conserved charges to remove the divergent part,

(∆φ)reg = ∆φ− 1

T1k

√

E2 −D2 −∑

i

P 2i = 2 cos−1(sin θ0) , (3.24)

which implies

sin θ0 = cos

(

(∆φ)reg2

)

. (3.25)

We can also regularize both the angular momenta as,

(J1)reg = J1 +2ω1(2ω

22 − 2α2 − c2ω2)

(c2 − 2ω2)(α2 − ω21)

√

E2 −∑

P 2i −D2

=2T1kω1(4ω

22 − 4α2 − 2c2ω2 − c2α)

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2cos θ0 (3.26)

and,

(J2)reg = J2 −2(c2α

2 + 2ω21ω2 − 2α2ω2 − c2ω

21)

(c2 − 2ω2)(α2 − ω21)

√

E2 −∑

P 2i −D2

=2T1kα(2ω2

1 − 2ω22 + c2ω2)− 2(2ω2

1ω2 − 2α2ω2 − c2ω21)

α√

(ω22 − ω2

1)4α2 − (c2 − 2ω2)2cos θ0 (3.27)

This regularisation for charges of a single-spike D1-string can be compared to similar con-

struction of F-strings in a background B field, considered in for example, [44]. It can

actually be shown that without the WZ term contribution, the angular momentum J1 and

J2 are finite as can also be seen in [19]. But with finite background flux, they diverge

and we have to regularise the charges to get a finite answer. These regularised angular

momenta can be easily found to satisfy the dispersion relation,

(J2)reg =

√

(J1)2reg + f1(λ) sin2

(

(∆φ)reg2

)

, (3.28)

where f1(λ) is a complicated function of various constants and winding numbers and N =√λ is the effective ’t Hooft coupling. However, if we choose c2 = 2ω2 − α, then the

complicated function reduces to f1(λ) =3λπ2 and the dispersion relation looks like,

(J2)reg =

√

(J1)2reg +3λ

π2sin2

(

(∆φ)reg2

)

, (3.29)

which matches exactly with the dispersion relation obtained by studying fundamental ro-

tating single-spike string solutions on NS5-branes [41].

– 12 –

JHEP04(2016)172

3.2 Giant Magnon

In this case we demand the situation where both ∂θ∂ξ1

and ∂φ2∂ξ1

diverge. Using these condi-

tions yield the relations between the constants α = ω1 and c3 = −2ω1. In this case also

the integration constant c2 remains undetermined. As in the previous case we will put the

value of c2 by hand to reduce the dispersion relation. Using these conditions we write down

the equations in the form,

∂φ2

∂ξ1=

(2ω22 − 2ω2

1 − c2ω2) sin2 θ

c2ω1 cos2 θ, (3.30)

and∂θ

∂ξ1=

sin θ√

sin2 θ − sin2 θ1cos θ sin θ1

, (3.31)

where the upper limit for θ is,

sin θ1 =c1c2ω1

√

(ω22 − ω2

1)(4c21ω

21 − c22κ

2 − 4κ2ω22 + 4c2κ2ω2)

. (3.32)

We can now easily integrate the θ equation to find the string profile as follows

ξ1 = cos−1

[

sin θ1sin θ

]

, (3.33)

which gives a finite range of ξ1 with |ξ1| ≤ π2 − θ1 as expected.

In this case we can construct the conserved charges in the usual sense,

E =2T1k(κ

2 − c21)(c2 − 2ω2) sin θ1c1c2ω1

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1,

Pi =2T1kνi(κ

2 − c21)(c2 − 2ω2) sin θ1c1c2κω1

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1,

D =2T1km(κ2 − c21)(c2 − 2ω2) sin θ1

c1c2κω1

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1. (3.34)

As in single-spike case before all these quantities diverge due to the divergent integral.

Among the angular momenta we can write,

J1 =2T1kω1(κ

2 − c21)(c2 − 2ω2) + 2c1κ(2ω22 − 2ω2

1 − c2ω2)c1c2κω1

∫ θ1

π2

sin θ1 sin θdθ

cos θ√

sin2 θ − sin2 θ1

− 2T1k2c1κ(2ω22−2ω2

1−c2ω2)+ω1(2c21ω2+c2κ

2−2κ2ω2)c1c2κω1

∫ θ1

π2

sin θ1 sin θ cos θdθ√

sin2 θ−sin2 θ1,

(3.35)

Now this J1 is divergent, but on the other hand,

J2 =2T1k[2c1c2κω1 − 2c21ω

21 − κ2ω2(c2 − 2ω2)]

c1c2κω1sin θ1 cos θ1 , (3.36)

– 13 –

JHEP04(2016)172

is finite. Also the angle deficit can be shown to be finite,

∆φ = −2 cos−1(sin θ1) . (3.37)

which implies sin θ1 = cos( (∆φ)2 ). If we define the divergent quantity,

E =ω1(κ

2 − c21)(c2 − 2ω2) + 2c1κ(2ω22 − 2ω2

1 − c2ω2)

α(κ2 − c21)(c2 − 2ω2)

√

E2 −D2 −∑

i

P 2i , (3.38)

then the quantity,

E − J1 =2T1k[2c1κ(2ω

21 − 2ω2

2 + c2ω2)− ω1(2c21ω2 + c2κ

2 − 2κ2ω2)]

c1c2κω1sin θ1 cos θ1 , (3.39)

is finite. This exactly adheres to the case of the dyonic giant magnon i.e. bound state of

J2 number of giant magnons. In that case we also have E and J1 divergent, but E − J1finite and J2 held fixed. We can write the final dispersion relation in the form,

E − J1 =

√

J22 + f2(λ) sin

2(∆φ

2

)

, (3.40)

where again we have used the definition of effective ’t Hooft coupling as before. Also f2(λ)

remains very complicated as in the previous case. But if we choose c2κ = 2κω2+ c1ω1 then

we can reduce it to f2(λ) =−3λπ2 and the dispersion relation will become,

E − J1 =

√

J22 − 3λ

π2sin2

(∆φ

2

)

. (3.41)

Once again one can see this relations exactly matches with the giant magnon dispersion

relation obtained in [41].

4 Rotating D1-strings on intersecting D5-D5’ branes

We shall now analyse the rotating D1-strings in the background of two stacks of D5-

branes intersecting on a line. More precisely, we have N1 D5-branes extended along

(x0, x1, x2, x3, x4, x5) direction and N2 D5-branes extended along (x0, x1, x6, x7, x8, x9) di-

rections, having exactly eight relatively transverse dimensions. These N1 and N2 have to

be large if we want the supergravity approximation to be valid, which constrain the regime

of the coupling constant where the solution is valid. For this reason, most of the literature

works with the S dual configuration [23]. Nonetheless, it is a very interesting solution

having the background of the form,

ds2 = (h1h2)−

12 (−dt2 + dx21) + h

−12

1 h122

5∑

i=2

dx2i + h121 h

−12

2

9∑

i=6

dx2i , (4.1)

together with a non-trivial dilaton,

e2φ = h−11 h−1

2 , (4.2)

– 14 –

JHEP04(2016)172

We have to remember that there are two sets of transverse spaces for the intersecting brane

configuration. For the set of branes having worldvolume lying on (x0, x1, x2, x3, x4, x5), the

transverse space is along (x6, x7, x8, x9). We can measure the transverse radial direction

in this case using r21 =∑9

i=6 x2i . Similarly for the second set of branes the transverse

direction is parameterised by r22 =∑5

i=2 x2i . The harmonic functions for these set of branes

are given by

h1 = 1 +k1r21

and h2 = 1 +k2r22

(4.3)

with k1 = gsN1l2s , k2 = gsN2l

2s and also

∑5i=2 dx

2i = dr22 + r22dΩ

22,

∑9i=6 dx

2i = dr21 + r21dΩ

21.

For this configuration it is natural to talk about dual 3-form RR fluxes which like the

previous section can actually be constructed from the volume form of the transverse spheres

(Ω1,Ω2) of the two sets of branes. Here we can parameterise the spheres as dΩ21 = dθ21 +

sin2 θ1dφ21+cos2 θ1dψ

21 and dΩ2

2 = dθ22+sin2 θ2dφ22+cos2 θ2dψ

22. Calculating the RR fluxes,

we get

Cφ1ψ1 = 2k1 sin2 θ1 , Cφ2ψ2 = 2k2 sin

2 θ2 . (4.4)

Now in the near-horizon (r1 → 0, r2 → 0), we can probe the region near the brane intersec-

tions. In this limit the harmonic functions approximate as h1 ≈ k1r21, h2 ≈ k2

r22. The metric

will reduce as,

ds2 =r1r2√k1k2

[

− dt2 + dx21 + k1

(

dr21r21

+ dΩ21

)

+ k2

(

dr22r22

+ dΩ22

)]

, (4.5)

The dilaton in this limit becomes,

e−φ =√

h1h2 =

√k1k2r1r2

. (4.6)

Before proceeding further we consider both the stacks contains same (large) number

of D5-branes i.e., N1 = N2 = N , which implies k1 = k2 = k. With these simplification the

metric, dilaton and magnetic dual RR fields become,

ds2 =r1r2k

[

− dt2 + dx21 + k

(

dr21r21

+ dΩ21

)

+ k

(

dr22r22

+ dΩ22

)]

,

e−φ =k

r1r2, Cφ1,ψ1 = 2k sin2 θ1 , Cφ2,ψ2 = 2k sin2 θ2 . (4.7)

Further, on rescaling t →√kt and x1 →

√kx1 and defining ρ1 = ln r1 and ρ2 = ln r2, the

metric and the dilaton becomes,

ds2 = eρ1eρ2[

− dt2 + dx21 + dρ21 + dθ21 + sin2 θ1dφ21 + cos2 θ1dψ

21

+ dρ22 + dθ22 + sin2 θ2dφ22 + cos2 θ2dψ

22

]

, e−φ = ke−ρ1e−ρ2 , (4.8)

– 15 –

JHEP04(2016)172

while the magnetic dual RR field remains unchanged. Now, the Lagrangian-density can be

written as,

L = −T1e−φ

√

− det Aαβ +T1

2ǫαβ∂αX

M∂βXNCMN

= −T1k[

[−∂0t∂1t+∂0x1∂1x1+∂0ρ1∂1ρ1+∂0θ1∂1θ1+sin2 θ1∂0φ1∂1φ1+cos2 θ1∂0ψ1∂1ψ1

+ ∂0ρ2∂1ρ2 + ∂0θ2∂1θ2 + sin2 θ2∂0φ2∂1φ2 + cos2 θ2∂0ψ2∂1ψ2]2 − [−(∂0t)

2 + (∂0x1)2

+ (∂0ρ1)2 + (∂0θ1)

2 + sin2 θ1(∂0φ1)2 + cos2 θ1(∂0ψ1)

2 + (∂0ρ2)2 + (∂0θ2)

2

+ sin2 θ2(∂0φ2)2+cos2 θ2(∂0ψ2)

2][−(∂1t)2+(∂1x2)

2+(∂1ρ1)2+(∂1θ1)

2+sin2 θ1(∂1φ1)2

+ cos2 θ1(∂1ψ1)2 + (∂1ρ2)

2 + (∂1θ2)2 + sin2 θ2(∂1φ2)

2 + cos2 θ2(∂1ψ2)2]]

12

+ 2T1k sin2 θ1(∂0φ1∂1ψ1 − ∂0ψ1∂1φ1) + 2T1k sin

2 θ2(∂0φ2∂1ψ2 − ∂0ψ2∂1φ2) . (4.9)

Before solving the Euler-Lagrange equations, we assume the following rotating string

ansatz,

t = κξ0 , x1 = vξ0 , ρi = miξ0 , θi = θi(ξ

1) ,

φi = νiξ0 + ξ1 , ψi = ωiξ

0 + ψi(ξ1) , i = 1, 2 . (4.10)

As we did in the last section also, these are basically generalizations of rotating F-string

solutions which can be shown to be consistent with the equations of motion. Now by

solving the equation of motion for t, we get,

κ(ν1 sin2 θ1 + ω1∂1ψ1 cos

2 θ1 + ν2 sin2 θ2 + ω2∂1ψ2 cos

2 θ2)√B

= c1 , (4.11)

where for convenience we have defined

B = [ν1 sin2 θ1 + ω1∂1ψ1 cos

2 θ1 + ν2 sin2 θ2 + ω2∂1ψ2 cos

2 θ2]2 − [−α2 + ν21 sin

2 θ1+

ω21 cos

2 θ1 + ν22 sin2 θ2 + ω2

2 cos2 θ2][(∂1θ1)

2 + (∂1θ2)2 + sin2 θ1

+ sin2 θ2 + cos2 θ1(∂1ψ1)2 + cos2 θ2(∂1ψ2)

2] (4.12)

and α2 = κ2 − v2 −m21 −m2

2 and c1 are just constants.

Now solving the equations of motion for φ1 and ψ1, we get,

sin2 θ1√B

[

ω1(ω1 − ν1∂1ψ1) cos2 θ1 + ν2(ν2 − ν1) sin

2 θ2

+ω2(ω2 − ν1∂1ψ2) cos2 θ2 − α2

]

− 2ω1 sin2 θ1 = c2 , (4.13)

cos2 θ1√B

[

ν1(ν1∂1ψ1 − ω1) sin2 θ1 + ν2(ν2∂1ψ1 − ω1) sin

2 θ2

+ω2(ω2∂1ψ1 − ω1∂1ψ2) cos2 θ2 − α2∂1ψ1

]

+ 2ν1 sin2 θ1 = c3 . (4.14)

– 16 –

JHEP04(2016)172

Again, solving for φ2 and ψ2, we get,

sin2 θ2√B

[

ν1(ν1 − ν2) sin2 θ1 + ω1(ω1 − ν2∂1ψ1) cos

2 θ1

+ω2(ω2 − ν2∂1ψ2) cos2 θ2 − α2

]

− 2ω2 sin2 θ2 = c4 , (4.15)

cos2 θ2√B

[

ν1(ν1∂1ψ2 − ω2) sin2 θ1 + ω1(ω1∂1ψ2 − ω2∂1ψ1) cos

2 θ1

+ν2(ν2∂1ψ2 − ω2) sin2 θ2 − α2∂1ψ2

]

+ 2ν2 sin2 θ2 = c5 . (4.16)

Solving (4.14) we get,

[c1ν21 sin

2 θ1 + c1ν22 sin

2 θ2 + c1ω22 cos

2 θ2 + 2κν1ω1 sin2 θ1 − c1α

2 − c3κω1]∂1ψ1 cos2 θ1

= (c1ω1 cos2 θ1 − 2κν1 sin

2 θ1 + c3κ)(ω2∂1ψ2 cos2 θ2 + ν1 sin

2 θ1 + ν2 sin2 θ2) . (4.17)

Again, solving (4.16) we get,

[c1ν21 sin

2 θ1 + c1ν22 sin

2 θ2 + c1ω21 cos

2 θ1 + 2κν2ω2 sin2 θ2 − c1α

2 − c5κω2]∂1ψ2 cos2 θ2

= (c1ω2 cos2 θ2 − 2κν2 sin

2 θ2 + c5κ)(ω1∂1ψ1 cos2 θ1 + ν1 sin

2 θ1 + ν2 sin2 θ2) . (4.18)

And finally from equation (4.11), we have,

(∂1θ1)2 + (∂1θ1)

2 =(c21 − κ2)(ν1 sin

2 θ1 + ω1∂1ψ1 cos2 θ1 + ν2 sin

2 θ2 + ω2∂1ψ2 cos2 θ2)

2

c21(−α2 + ν21 sin2 θ1 + ω2

1 cos2 θ1 + ν22 sin

2 θ2 + ω22 cos

2 θ2)

− sin2 θ1 − sin2 θ2 − cos2 θ1(∂1ψ1)2 − cos2 θ2(∂1ψ2)

2 . (4.19)

Equation (4.19) have two variables ∂1θ1 and ∂1θ2, it is quite difficult to find the so-

lutions of both ∂1θ1 and ∂1θ2 simultaneously from the form of the equation (4.19). For

simplicity we will now consider θ1 = θ2 = θ, which will confine the string on both the

spheres but with restriction that both the sphere will have the same value of θ. This will

make our equations more tractable.

Also, we define the following quantities for our convenience,

a = c1(ν21 + ν22) sin

2 θ − c1α2 , b = c1ω

22 cos

2 θ + 2κν1ω1 sin2 θ − c3κω1 ,

c = c1ω21 cos

2 θ + 2κν2ω2 sin2 θ − c5κω2 , d = (ν1 + ν2) sin

2 θ ,

e = c1ω1 cos2 θ − 2κν1 sin

2 θ + c3κ , f = c1ω2 cos2 θ − 2κν2 sin

2 θ + c5κ , (4.20)

then equation (4.17) and (4.18) can be written as,

(a+ b)∂1ψ1 cos2 θ = e(ω2∂1ψ2 cos

2 θ + d) ,

(a+ c)∂1ψ2 cos2 θ = f(ω1∂1ψ1 cos

2 θ + d) . (4.21)

By solving the above equations we get,

∂1ψ1 =ed(a+ c+ ω2f)

cos2 θ[(a+ b)(a+ c)− ω1ω2ef ],

∂1ψ2 =fd(a+ b+ ω1e)

cos2 θ[(a+ b)(a+ c)− ω1ω2ef ]. (4.22)

– 17 –

JHEP04(2016)172

Again using these results equation (4.19) reduces to,

(∂1θ)2 =

(κ2 − c21)

2c21

[(ν1 + ν2) sin2 θ + (ω1∂1ψ1 + ω2∂2ψ2) cos

2 θ]2

[α2 − (ν21 + ν22) sin2 θ − (ω2

1 + ω22) cos

2 θ]

− sin2 θ − 1

2[(∂1ψ1)

2 + (∂1ψ2)2] cos2 θ . (4.23)

The above set of equations of motion appear to be generalizations of the ones discussed in

the last section for single stack of D5 branes. We will exactly follow the way we found out

the single spike and giant magnon solutions for the last section. Namely we will impose

the similar kind of conditions on ∂1θ, ∂1ψ1 and ∂1ψ2 and define the conserved charges in

each case separately.

4.1 Single Spike-like solution

For a generalised spike solution, we impose the condition on the derivatives

∂1θ → 0 as θ → π

2, (4.24)

Implementing the conditions, from equation (4.23), we get,

c1 =κ(ν1 + ν2)

α1, where α1 =

√

2α2 − (ν1 − ν2)2 . (4.25)

Again, follwing our earlier discussion we will demand that the derivatives ∂1ψ1 and ∂2ψ2

have to be regular in the limit θ → π2 . This leads us to the relations between the constants

c3 = 2ν1 and c5 = 2ν2 respectively. Using these relations, we get,

∂1ψ1 =(ν1 + ν2)(c1ω1 + 2κν1) sin

2 θ

c1(ν21 + ν22) sin2 θ − 2κα2 cos2 θ − c1α2

,

∂1ψ2 =(ν1 + ν2)(c1ω2 + 2κν2) sin

2 θ

c1(ν21 + ν22) sin2 θ − 2κα2 cos2 θ − c1α2

, (4.26)

where α2 = ν1ω1 + ν2ω2. Similarly, equation (4.23) reduces to,

∂1θ =

√

a12

κ sin θ cos θ√

sin2 θ − sin2 θ0

α1[c1(ν21 + ν22) sin2 θ − 2κα2 cos2 θ − c1α2]

, (4.27)

where

sin θ0 =2α1α2 + (ν1 + ν2)α

2

√

a12

(4.28)

and

a1 = 2α2(ν21 + ν22)(ν1 + ν2)2 + 8α2

1α22 + 8α1α2(ν1 + ν2)(ν

21 + ν22)− 4α1α2(ν1 + ν2)

3

− α21(ν1 + ν2)(ω

21 + ω2

2 + 4ν21 + ν22). (4.29)

Now, like the previous section we can solve for the string profile by integrating the θ

equation. We will not present the cumbersome expressions here for brevity.

– 18 –

JHEP04(2016)172

Let us now define the conserved charges corresponding to the various isometries as

given by,

E = −2T1

∫

∂L∂(∂0t)

dθ

∂1θ=

4T1kκ(ν1 + ν2)(ν21 + ν22 − α2)

α1

√

a12

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0,

P = 2T1

∫

∂L∂(∂0x1)

dθ

∂1θ=

4T1kv(ν1 + ν2)(ν21 + ν22 − α2)

α1

√

a12

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0,

D1 = 2T1

∫

∂L∂(∂0ρ1)

dθ

∂1θ=

4T1km1(ν1 + ν2)(ν21 + ν22 − α2)

α1

√

a12

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0,

D2 = 2T1

∫

∂L∂(∂0ρ2)

dθ

∂1θ=

4T1km2(ν1 + ν2)(ν21 + ν22 − α2)

α1

√

a12

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0.

(4.30)

Like the previous section, the charges D1 and D2 are associated with shifts in ln r1 and

ln r2, which again are symmetries of the D1 string action. All these above quantities can

be shown to diverge. But, using these expressions we can define a new divergent quantity,

√

E2 − P 2 −D21 −D2

2 =4T1kα(ν1 + ν2)(ν

21 + ν22 − α2)

α1

√

a12

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0.

(4.31)

The angle deficit ∆φ = 2∫

dθ∂1θ

is given by,

∆φ =2α1

κ√

h12

[

c1(ν21 + ν22 − α2)

∫ π2

θ0

sin θdθ

cos θ√

sin2 θ − sin2 θ0

−(c1α2 + 2κα2)

∫ π2

θ0

sin θ cos θdθ√

sin2 θ − sin2 θ0

]

. (4.32)

This is also divergent, but we can regularize it by removing the divergent part,

(∆φ)reg = ∆φ− α1

2T1kα

√

E2 − P 2 −D21 −D2

2 = − cos−1(sin θ0) , (4.33)

which implies sin θ0 = cos((∆φ)reg

2 ). Angular momenta J1 = 2T1

∫

∂L∂(∂1φ1)

dθ∂1θ

and

J2 = 2T1

∫

∂L∂(∂1φ2)

dθ∂1θ

also diverge. Regularised values of J1 and J2 are given by,

(J1)reg = J1 −2α1(ν1 + ν2)ω1 + 4α2

1ν1 + (ν1 − ν2)(ν21 + ν22 − α2)

2α(ν21 + ν22 − α2)

√

E2 − P 2 −D21 −D2

2

=2T1k(ν1 + ν2)

α1

√

a12

[2α1ν2(ω2 − ω1) + 2ν1(α2 − 2α2

1)− (ν1 − ν2)(ν21 + ν22)] cos θ0 ,

(J2)reg = J2 −2α1(ν1 + ν2)ω2 + 4α2

1ν1 + (ν2 − ν1)(ν21 + ν22 − α2)

2α(ν21 + ν22 − α2)

√

E2 − P 2 −D21 −D2

2

=2T1k(ν1 + ν2)

α1

√

a12

[2α1ν1(ω1 − ω2) + 2ν2(α2 − 2α2

1) + (ν1 − ν2)(ν21 + ν22)] cos θ0 .

(4.34)

– 19 –

JHEP04(2016)172

Again the angular momenta K1 = 2T1

∫

∂L∂(∂1ψ1)

dθ∂1θ

and K2 = 2T1

∫

∂L∂(∂1ψ2)

dθ∂1θ

also

diverges. Regularised K1 and K2 are given by,

(K1)reg = K1 +α1

α

√

E2 − P 2 −D21 −D2

2

=2T1k√

a12

[4α1α2 − (ν1 + ν2)ω1α1 − 2ν22(ν21 − ν22)] cos θ0 ,

(K2)reg = K2 −α1

α

√

E2 − P 2 −D21 −D2

2

=2T1k√

a12

[4α1α2 − (ν1 + ν2)ω2α1 − 2ν21(ν22 − ν21)] cos θ0 , (4.35)

Now, defining Jreg = (J1)reg + (J2)reg and Kreg = (K1)reg − (K2)reg, we find that they

satisfy a generalized dispersion relation of form,

Jreg =

√

K2reg + f3(λ) sin

2

(

(∆φ)reg2

)

, (4.36)

where f3(λ) =2λπ2

(ν1+ν2)2

a1α21

[2α1(ω2−ω1)(ν2−ν1)+2(ν1+ν2)(α2−2α2

1)2−α21(ω2−ω1)α1+

(ν22 − ν21)2].

4.2 Giant Magnon-like solution

Here we use the opposite condition on the equations of motion, i.e. we impose that ∂1θ,

∂1ψ1 and ∂1ψ2 diverges as θ → π2 . Imposing the first one, we get the following condition,

α2 = ν21 + ν22 (4.37)

we can not determine the values of c3 and c5 by this condition alone. As we have found in

the previous cases, we put c3 = 2ν1 and c5 = 2ν2 by hand. Using these values of constants,

we get,

∂1ψ1 = −(ν1 + ν2)(c1ω1 + 2κν1) sin2 θ

β cos2 θ,

∂1ψ2 = −(ν1 + ν2)(c1ω2 + 2κν2) sin2 θ

β cos2 θ, (4.38)

where β = c1(ν21 + ν22) + 2κα2. Also, equation (4.23) reduces to,

∂1θ =sin θ

√

sin2 θ − sin2 θ1sin θ1 cos θ

, (4.39)

where sin θ1 =β√a22

and a2 = 2β2 − (ν1 + ν2)2[(3κ2 + c21)(ν

21 + ν22) + κ2(ω2

1 +ω22 +4c2κα2)].

– 20 –

JHEP04(2016)172

In this case, the conserved charges are given by,

E =2T1k(ν1 + ν2)(c

21 − κ2)

√

a22

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1,

P =2T1kv(ν1 + ν2)(c

21 − κ2)

κ√

a22

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1,

D1 =2T1km1(ν1 + ν2)(c

21 − κ2)

κ√

a22

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1,

D2 =2T1km2(ν1 + ν2)(c

21 − κ2)

κ√

a22

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1. (4.40)

All these quantities diverge and we again define the quantity,

√

E2 − P 2 −D21 −D2

2 =2T1kα(ν1 + ν2)(c

21 − κ2)

κ√

a22

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1. (4.41)

The angle deficit is finite and is given by,

∆φ = 2 sin θ1

∫ θ1

π2

cos θdθ

sin θ√

sin2 θ − sin2 θ1= 2 cos−1(sin θ1) , (4.42)

which implies sin θ1 = cos(∆φ2 ). Again, the angular momenta J1 and J2 are given by,

J1 =2T1k

κ√

a22

[

[c1β + (ν1 + ν2)2κ(c1ω1 + 2κν1)− ν1(c21 − κ2)]

∫ θ1

π2

sin θ cos θdθ√

sin2 θ − sin2 θ1

+ (ν1 + ν2)[ν1(c21 − κ2)− 2κ(c1ω1 + 2κν1)]

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1

]

, (4.43)

J2 =2T1k

κ√

a22

[

[c1β + (ν1 + ν2)2κ(c1ω2 + 2κν2)− ν2(c21 − κ2)]

∫ θ1

π2

sin θ cos θdθ√

sin2 θ − sin2 θ1

+ (ν1 + ν2)[ν2(c21 − κ2)− 2κ(c1ω2 + 2κν2)]

∫ θ1

π2

sin θdθ

cos θ√

sin2 θ − sin2 θ1

]

, (4.44)

But, the angular momenta K1 and K2 are finite and are given by,

K1 =2T1k√

a22

[(ν1 + ν2)(ω1κ+ 2c1ν1) + 2β] cos θ1 ,

K2 =2T1k√

a22

[(ν1 + ν2)(ω2κ+ 2c1ν2) + 2β] cos θ1 , (4.45)

Now, defining J = J1 + J2, K = K1 −K2, and

E =(ν1 − ν2)(c

21 − 5κ2) + 2c1κ(ω1 − ω2)

α(c21 − κ2)

√

E2 − P 2 −D21 −D2

2 , (4.46)

we find that they satisfy a dispersion relation of form,

E − J =

√

K2 + f4(λ) sin2(∆φ

2

)

, (4.47)

– 21 –

JHEP04(2016)172

where f4(λ)=2λπ2

(ν1+ν2)2

a2κ2[(ν2−ν1)(c

21−5κ2)+2c1κ(ω1−ω2)2−κ2κ(ω1−ω2)+2c1(ν1+ν2)2].

Again, it is noteworthy that the same form of generalised giant magnon solution was found

for a rotating F-string in [42].

5 Conclusions

In this paper, we have studied various solutions of the D-string equations of motion in

various curved backgrounds. First we have studied string equations of motion, of a bound

state of oscillating D1 strings and F-strings with non-trivial gauge field on the D1 world-

volume, in the recently found AdS3 background with mixed fluxes. One of the interesting

outcome of this solution is that the periodically expanding and contracting (1, n) string

has a possibility of reaching the boundary of AdS3 in finite time in contrast to the probe

D-string motion in the WZW model with only NS-NS fluxes [16]. Further we have studied

the DBI equations of motion of the D1-string in D5-brane background and found out two

classes of solutions corresponding to the giant magnon and single spike solutions of the

D-string. Also, we have studied the rigidly rotating D1-string solution in the intersecting

D5-brane background and compared our results with the existing S-dual configurations.

These solutions are basically generalisations of the usual single spike and giant magnon

solutions for a D1-string. However, the physical interpretation of such string states remain

elusive from us as of now.

Acknowledgments

It is a pleasure to thank Kamal L. Panigrahi for discussions and important suggestions

on the manuscript. The authors are indebted to S. Pratik Khastgir for various fruitful

discussions. AB would like to thank Pulastya Parekh and Abhishake Sadhukhan for their

comments. He would also like to thank the organisers of National String Meeting 2015 for

kind hospitality at IISER Mohali, where a part of this work was done.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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